By considering E the infinite dimensional Grassmann algebra over a field of characteristic zero and the T-prime algebra M = M
2
(E) with the ℤ
2
-grading
, in this article we describe the space of ...the graded central polynomials modulo the ideal of the graded identities of M.
The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic
p
>
2
little is known about them. In previous papers we discussed some sharp ...differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras
M
1
,
1
(
E
)
and
E
⊗
E
are not PI equivalent in characteristic
p
>
2
. Furthermore we show that the following algebras are not PI equivalent in positive characteristic:
M
a
,
b
(
E
)
⊗
E
and
M
a
+
b
(
E
)
;
M
a
,
b
(
E
)
⊗
E
and
M
c
,
d
(
E
)
⊗
E
when
a
+
b
=
c
+
d
,
a
⩾
b
,
c
⩾
d
and
a
≠
c
; and finally,
M
1
,
1
(
E
)
⊗
M
1
,
1
(
E
)
and
M
2
,
2
(
E
)
. Here
E stands for the infinite-dimensional Grassmann algebra with 1, and
M
a
,
b
(
E
)
is the subalgebra of
M
a
+
b
(
E
)
of the block matrices with blocks
a
×
a
and
b
×
b
on the main diagonal with entries from
E
0
, and off-diagonal entries from
E
1
;
E
=
E
0
⊕
E
1
is the natural grading on
E.
Let
K be a field,
char
K
=
0
. We study the polynomial identities satisfied by
Z
2
-graded tensor products of T-prime algebras. Regev and Seeman proved that in a series of cases such tensor products ...are PI equivalent to T-prime algebras; they conjectured that this is always the case. We deal here with the remaining cases and thus confirm Regev and Seeman's conjecture. For some “small” algebras we can remove the restriction on the characteristic of the base field, and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual (ungraded) tensor products. Finally we consider
β-graded tensor products (also called commutation factors) and their identities. We show that Regev's
A
⊗
B
theorem holds for
β-graded tensor products whenever the gradings are by finite abelian groups. Furthermore we study the PI equivalence of
β-graded tensor products of T-prime algebras.
Let K be a field, char K = 0, and let E = E
0
⊕ E
1
be the Grassmann algebra of infinite dimension over K, equipped with its natural ℤ
2
-grading. If G is a finite abelian group and R = ⨁
g∈G
R
(g)
...is a G-graded K-algebra, then the algebra R⊗ E can be G × ℤ
2
-graded by setting (R⊗ E)
(g, i)
= R
(g)
⊗ E
i
. In this article we describe the graded central polynomials for the T-prime algebras M
n
(E)≅ M
n
(K)⊗ E. As a corollary we obtain the graded central polynomials for the algebras M
a, b
(E)⊗ E. As an application, we determine the ℤ
2
-graded identities and central polynomials for E⊗ E.
In this paper we study the graded identities satisfied by the superalgebras
M
a
,
b
over the Grassmann algebra and by their tensor products. These algebras play a crucial role in the theory developed ...by A. Kemer that led to the solution of the long standing Specht problem. It is well known that over a field of characteristic 0, the algebras
M
pr
+
qs
,
ps
+
qr
and
M
p
,
q
⊗
M
r
,
s
satisfy the same
ordinary polynomial identities. By means of describing the corresponding graded identities we prove that the T-ideal of the former algebra is contained in the T-ideal of the latter. Furthermore the inclusion is proper at least in case
(
r
,
s
)
=
(
1
,
1
)
. Finally we deal with the graded identities satisfied by algebras of type
M
2
n
-
1
,
2
n
-
1
and relate these graded identities to the ones of tensor powers of the Grassmann algebra. Our proofs are combinatorial and rely on the relationship between graded and ordinary identities as well as on appropriate models for the corresponding relatively free graded algebras.
Let
K
be an infinite integral domain, and let
A
=
M
2
(
K
) be the matrix algebra of order two over
K
. The algebra
A
can be given a natural
-grading by assuming that the diagonal matrices are the ...0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of
A
. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when
K
is an infinite field of characteristic 0 or
p
> 2. Our proofs are characteristic-free so they work when
K
is an infinite field, char
K
= 2. Thus we describe finite bases of the graded identities and graded central polynomials for
M
2
(
K
) in this case as well.
The algebras M
a, b
(E) ⊗ E and M
a+b
(E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor ...product theorem. It was first proved by Kemer in 1984-1987 (see Kemer
1991
); other proofs of it were given by Regev (
1990
), and in several particular cases, by Di Vincenzo (
1992
), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M
1, 1
(E) ⊗ E and M
2
(E) when the base field is infinite and of characteristic p > 2. The algebra M
a, a
(E) ⊗ E satisfies certain graded identities that are not satisfied by M
2a
(E). In another paper we proved that the algebras M
1, 1
(E) and E ⊗ E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.
In this paper we study tensor products of
T-prime
T-ideals over infinite fields. The behaviour of these tensor products over a field of characteristic 0 was described by Kemer. First we show, using ...methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the tensor product theorem fails for the
T-ideals of the algebras
M
1,1(
E) and
E⊗
E where
E is the infinite-dimensional Grassmann algebra;
M
1,1(
E) consists of the 2×2 matrices over
E having even (i.e., central) elements of
E, and the other diagonal consisting of odd (anticommuting) elements of
E. Note that these proofs do not depend on the structure theory of
T-ideals but are “elementary” ones. All this comes to show once more that the structure theory of
T-ideals is essentially about the multilinear polynomial identities.
Let be a field of characteristic zero, and let us consider the matrix algebra M
2
( ) endowed with the ℤ
2
-grading ( e
11
⊕ e
22
) ⊕ ( e
12
⊕ e
21
). We define two superalgebras, ℛ
p
and
q
, ...where p and q are positive integers. We show that if is a proper subvariety of the variety generated by the superalgebra M
2
( ), then the even-proper part of the T
2
-ideal of graded polynomial identities of asymptotically coincides with the even-proper part of the graded polynomial identities of the variety generated by the superalgebra ℛ
p
⊕
q
. This description also affords an even-asymptotic description of the proper subvarieties of the variety generated by the superalgebra M
1,1
(E) as even-asymptotically coinciding with the T
2
-ideal of the variety generated by the Grassmann envelopes G(ℛ
p
) and G(
q
). Moreover, the following general fact is established. If two varieties of superalgebras are even-asymptotically equivalent, then they are asymptotically equivalent, and they have the same PI-exponent.
Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z-gradings on E of full support. Using methods of elementary ...number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that their Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of E.
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